Problem: Dingane has been observing a certain stock for the last few years and he sees that it can be modeled as a function $S(t)$ of time $t$ (in days) using a sinusoidal expression of the form $a\cdot\sin(b\cdot t)+d$. On day $t=0$, the stock is at its average value of ${\$}3.47$ per share, but $91.25$ days later, its value is down to its minimum of $\$1.97$. Find $S(t)$. $\textit{t}$ should be in radians. $S(t) = $
Answer: The strategy First, we should convert the given information about the real-world context into mathematical terms of the sinusoidal function and its graph. Then, we should use the given information to find the amplitude, midline, and period of the function's graph. Finally, we should find $a$, $b$, and $d$ in the expression $a\sin(b\cdot t)+d$ by considering the features we found. Converting the given information into mathematical terms At $t=0$, the stock is valued at $\$3.47$ per share. This means the graph of the function passes through $(0,3.47)$. We are given that this is the average value, which corresponds to the midline of the graph. $91.25$ days later (which means $t=91.25$ ) the stock's value is $\$1.97$. This corresponds to the point $(91.25,1.97)$. We are given that this is the minimum value, which corresponds to a minimum point of the graph. In conclusion, the graph intersects its midline at $(0,3.47)$ and then has a minimum point at $(91.25,1.97)$. Determining the amplitude, midline, and period The midline intersection is at $y={3.47}$, so this is the midline. The minimum point is $1.5$ units below the midline, so the amplitude is ${1.5}$. The minimum point is $91.25$ units to the right of the midline intersection, so the period is $4\cdot 91.25={365}$. [Why did we multiply by 4?] Determining the parameters in $a\sin(b\cdot t)+d$ Since the midline intersection at $t=0$ is followed by a minimum point, we know that $a<0$. [How do we know that?] The amplitude is ${1.5}$, so $|a|={1.5}$. Since $a<0$, we can conclude that $a=-1.5$. The midline is $y={3.47}$, so $d=3.47$. The period is ${365}$, so $b=\dfrac{2\pi}{{365}}$. The answer $S(t)=-1.5\sin\left(\dfrac{2\pi}{365}t\right)+3.47$